A THEOREM IN ELLIPTIC FUNCTIONS. 
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that is, (z 2 - w 2 ) (1 - fey 2 ) - (x 2 - y)(l _ fe 2 w 2 ) 
= 2 {¿cm; Vl ~y\ 1 -Pif. 1 - ¿ 2 .1 - fe 2 - y Z Vl - x 2 . 1 - fe 2 .1 - w 2 . 1 - fe 2 }. 
Rationalising, we obtain, as mentioned above, an equation containing only the squares 
x 2 , y 2 , z 2 , vf-; it therefore is of a degree twice that of the equation containing 
the product xyzw. I worked out in this way the equation in (x 2 , y 2 , z 2 , tu 2 ), but the 
calculation was lost, and the easier way of obtaining it is obviously by means of the 
equation involving xyzw. 
We have, by the theorem, 
that is, 
and then, writing 
— k' 2 xyzw 
+ Vl — x 2 .1 — y 2 .1 — z 2 .1 — w 2 
1 £'2 
— j- Vl — fe 2 .1 — k 2 y 2 . 1 — fe 2 .1 — k 2 w 2 — — , 
k 2 J p 
k' 2 (1 — k 2 xyzw) = k 2 Vl — a? . 1 — y 2 . 1 — z 2 .1 — w 2 
— Vl — fe 2 . 1 — k 2 y 2 .1 — Pz 2 .1 — k 2 w 2 ; 
P = x 2 + y 2 + z 2 + w 2 , 
Q = x 2 y 2 + x 2 z 2 + x 2 w 2 + y 2 z 2 + y 2 w 2 + z : w 2 , 
R = x 2 y 2 z 2 + x 2 y 2 w 2 + x 2 z 2 w 2 + y 2 z 2 w 2 , 
S =x 2 y 2 z 2 w 2 , 
and using aJS to denote the rational function xyziv, we have 
Af 4 (l -2fc*JS + k*S) 
= t(l-P + Q-R + S) 
+1 - k 2 P + k*Q - k 6 R + PS 
— 2k 2 V(1 — P + Q — R + S) (1 — k 2 P + PQ — PR + k 8 S); 
or, if for a moment the radical is called \/A, then the factor k 2 divides out, and 
the equation becomes 
2 VA = 2 - (1 + A; 2 ) P + 2k 2 Q ~(k 2 + P)R + 2PS + 2&' 4 PS, 
whence 
4(1_p + Q-R + S) (1 -PP + PQ-PR + PS) 
_ |2 - (1 + P) P + 2PQ - (k 2 + P)R+ 2PS\ 2 - 
= - 2P PS [2 - (1 + P) P + 2PQ - (P + P) R + 2fe}. 
The factor P 4 divides out; omitting it, we have 
4Q _ ps _ 4 (i + p) R + 1QPS + 2PPR - 4 (P + P) PS - PR 2 + WQS 
= ~2PS{2-(1+ P) P + 2 PQ - (P + P)R + 2 PS], 
or, as this may also be written, 
j(_ P 2 + 4Q - 4P) + P (- 4>R + 2PR + 16S- 4P>8) -f P (- R 2 + 4QS - PS)} 
= -2 PS[2-P + P(-P + 2Q-R)+P(-R+2S)}, 
which is the required rational equation involving the product of the sn s.